It is an undisputed fact that correlation (or regression) between any two time series is valid if, and only if, the series are stationary, i.e., if the mean of the series is constant throughout the time course and thus independent of the location of the time instant where it is calculated along the series (Priestley,

1981; Box et al.,

2008; Jenkins and Watts,

1968; Bartlett,

1978). If this assumption does not hold, i.e., if the series are nonstationary, then spurious correlations are obtained. This issue had been recognized in the early twentieth century (Yule,

1926) but was brought forcefully to the foreground in the 1970s in the field of econometrics (Granger and Newbold,

1977). It ultimately led to a Nobel Prize awarded in 2003 to Clive Granger for (among other achievements) demonstrating that “the statistical methods used for stationary time series could yield wholly misleading results when applied to the analysis of nonstationary data.” (From Nobel Prize citation:

http://nobelprize.org/nobel_prizes/economics/laureates/2003/press.html). Granger also discussed this issue in his Nobel Prize Lecture (Granger,

2004). The solution to the problem is to render the series stationary which can be achieved by differencing (i.e., taking successive differences in the series) or by the application of various detrending procedures. The hallmark of nonstationarity is the presence of autocorrelations in the series which can be detected by calculating the autocorrelation and partial autocorrelation functions (ACF and PACF, respectively). Suitable detrending will remove autocorrelations due to trends. However, a suitably detrended series may still show significant autocorrelations due to the presence of other processes, typically AR and MA processes (Box and Jenkins,

1970). The presence of an AR process indicates a dependence of a given value on previous values (irrespective of trend), whereas the presence of a MA process indicates a dependence of a given value on the variation (“random shock”) of previous values. Such dependencies need to be removed before correlation or regression analyzes are performed to ensure that the correlation obtained is truly due to the relation between the two time series and does not simply reflect, or is contaminated by, the time history of the series themselves. In a similar approach, as that applied for detrending, AR and MA dependencies are detected and removed. When this preprocessing is complete, i.e., when trends are removed and AR and/or MA dependencies eliminated, the resulting series are stationary and nonautocorrelated, which means that they are now ready to be correlated. Because there are no dependencies on previous values, the new “clean” series are called “innovations.” The standard way to accomplish this task consists of a three stage process (Box et al.,

2008): (1) identify the sources of dependencies in the series (“model identification”), (2) calculate the coefficients for these dependencies (“model estimation”), and (3) take the residuals (innovations). As the final diagnostic check, the innovations time series should be stationary and non-autocorrelated. As mentioned above, there are three major sources of dependencies, namely trends (“Integrated” series), AR, and MA processes. The three stage process above used to identify, estimate and remove these dependencies is called ARIMA, from the initials (AR Integrated MA) (Box and Jenkins,

1970). Since the innovations series are essentially white noise, the data preprocessing above is called “prewhitening,” a term coined by John Tukey in 1956 (Press and Tukey,

1956) in the context of spectral analysis of time series. Detailed exposition, discussion and remedies for these problems can be found in any textbook on time series analysis (e.g., Priestley,

1981; Box et al.,

2008), as detailed in a previous paper (Christova et al.,

2011). It is unclear why this fundamental point has been so pervasively ignored in functional neuroimaging, while it has been commonplace in other fields, notably econometrics, for at least the past three decades. An obvious reason is that correlating autocorrelated time series requires special care of which the general neuroscientist is unaware. For example, a fundamental assumption of least squares linear regression is that the error terms be independent, and violation of this assumption invalidates statistical testing of the significance of regression slope, since its standard error is wrongly estimated. Although this is standard textbook knowledge (see, e.g., Snedecor and Cochran,

1989), it is not generally appreciated that regressing (or correlating) autocorrelated time series violates exactly this fundamental assumption, leading to inflated correlations (Pierce,

1979). In fact, in two early influential papers on fMRI data analysis, it was explicitly assumed that the errors in regression analysis between time series are independent, an erroneous assumption (Friston et al.,

1995; Worsley and Friston,

1995). The detrimental effect of non-independent errors in regression analysis has been lucidly exposed by Box et al. (

1978). It is interesting that the uncertainty as to how to correlate time series is clearly present in most recent textbooks of fMRI data analysis (Ashby,

2011; Poldrack et al.,

2011).